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A copolymer is a chain of repetitive units (monomers) that
are almost identical, but they differ in their degree of
affinity for certain solvents. This difference leads to striking
phenomena when the polymer fluctuates
in a non-homogeneous medium, for example made up by two solvents
separated by an interface.
One may observe, for exmple, the localization of the polymer at the
interface between the two solvents.
Much of the literature on the subject focuses on the most basic model
based on the simple symmetric random walk on the integers, but
E. Bolthausen and F. den Hollander (AP 1997) pointed out
the convergence of the (rescaled) free energy of such a discrete model
toward
the free energy of a continuum model, based on Brownian motion,
in the limit of weak polymer-solvent coupling. This result is
remarkable because it strongly suggests
a universal feature for copolymer models. In this work we prove that
this is indeed the case. More precisely,
we determine the weak coupling limit for a general class of discrete
copolymer models, obtaining as limits
a one-parameter (alpha in (0,1)) family of continuum models, based on
alpha-stable regenerative sets.